if you got a NOC you could even start a new job without leaving qatar for a minute.. so definitely you can come back in 3 months or less..
new passport won't help. they check you by your fingerprints, not by your passport..and hey.. it's not nice to think about the way of fooling the government of State of Qatar, let alone asking about it on the public forum !

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“You become responsible forever for what you have tamed”. Antoine de Saint-Exupéry